Note that there is a discrepancy between the two mixing times we get from Theorems REF 
and REF .
This has to do with how we use spectral independence, i.e., once this has been established, to bound the mixing
time of Glauber dynamics. In that respect, we make a direct use of results from [1]}, [2]}
and hence, the discrepancy comes from the fact that these two papers derive different bounds on the mixing time.
In the related literature there is also the work in [3]} which can be considered
for the unbounded case. However, it seems that the result in [3]}
requires from the Gibbs distribution to satisfy a strong condition called Gibbs uniqueness. The range of the parameters
we consider for the Gibbs distribution in Theorems REF  and
REF  is beyond this region.
